Beyond the Modifiable Areal Unit Problem!
Why fractals are so useful for Geographers?
In the early eighties, fractals were everywhere, from the shape of clouds to the spatial organisation of galaxies. Now scientists are more nuanced. Fractal is still a very important area of research, but there is no need for the world to be fractal in order to use fractal analysis.
Before we discuss the usefulness of fractals for geographers, it might be relevant to introduce them. Fractals were invented - discovered - by Benoît Mandelbrot, a French-American mathematician who believed that the world was not smooth, but rough. No matter how you look at it, every part of an object looks like the whole object. This is one way to describe what a fractal is. Another way is to say that a fractal has lots of detail at each scale. The two points of view are not opposite, as we can see.
The dimension of information
Fortunately, fractal theory isn’t just a way of making fancy pictures, it’s also a toolbox for measuring how the scales are related. Benoît Mandelbrot uses fractal dimensions to do this. There are a plethora of fractal dimensions, from the very theoretical to the more practical uses in Physics.
In this article we focus on a specific dimension called the information dimension. It is based on the Shannon entropy, which is the number of Boolean questions we have to ask to find an element in an ensemble.
And what about geography…
In geography, the role of scale is a major concern. It has been theorised under the concept of the Modifiable Areal Unit Problem (MAUP). The resulting summary values (e.g., totals, rates, proportions, densities) are influenced by both the shape and scale of the aggregation unit.